Generating function for the Catalan numbers

Question

I know that generating function $f(x)$ for the Catalan numbers is \begin{equation} f(x)=\cfrac{1\pm \sqrt{1-4x}}{2x}\ . \end{equation}

It is often said that we should choose \begin{equation} f(x)=\cfrac{1- \sqrt{1-4x}}{2x} \end{equation}

because $f(x)$ should be continuous at $x=0$, but I can't understand why $f(x)$ should be continuous.

What is the problem if $f(x)$ is not continuous at $x=0$ ?

Answer 1

We choose the negative sign in \begin{align*} f(x)=\frac{1\color{blue}{\pm} \sqrt{1-4x}}{2x} \end{align*} since we want to expand $f$ in a power series.

According to the binomial series expansion we have for $|x|<\frac{1}{4}$ the following representation at $x=0$ \begin{align*} \sqrt{1-4x}&=\sum_{n=0}^\infty \binom{\frac{1}{2}}{n}(-4x)^n\\ &=1-2x-2x^2-4x^3-10x^4-\cdots \end{align*} so that \begin{align*} \frac{1+\sqrt{1-4x}}{2x}=\color{blue}{\frac{1}{x}}-1-x-2x^2-5x^3-14x^4-\cdots\tag{1} \end{align*} whereas \begin{align*} \frac{1-\sqrt{1-4x}}{2x}=1+x+2x^2+5x^3+14x^4+\cdots\tag{2} \end{align*}

Note the latter (2) is a power series, while the former (1) is not a power series.